Download Measuring fractals by infinite and infinitesimal numbers

MEASURING FRACTALS BY INFINITE AND INFINITESIMAL
NUMBERS
Yaroslav D. Sergeyev
DEIS, University of Calabria, Via P. Bucci, Cubo 42-C, 87036 Rende (CS), Italy,
N.I. Lobachevsky State University, Nizhni Novgorod, Russia, and
Institute of High Performance Computing and Networking of the National Research Council of Italy
http://wwwinfo.deis.unical.it/∼yaro
e-mail: [email protected]
Abstract. Traditional mathematical tools used for analysis of fractals allow one to distinguish results
of self-similarity processes after a finite number of iterations. For example, the result of procedure of
construction of Cantor’s set after two steps is different from that obtained after three steps. However, we
are not able to make such a distinction at infinity. It is shown in this paper that infinite and infinitesimal
numbers proposed recently allow one to measure results of fractal processes at different iterations at
infinity too. First, the new technique is used to measure at infinity sets being results of Cantor’s procedure. Second, it is applied to calculate the lengths of polygonal geometric spirals at different points of
infinity.
1. I NTRODUCTION
During last decades fractals have been intensively studied and applied in various
fields (see, for instance, [4, 11, 5, 7, 12, 20]). However, their mathematical analysis
(except, of course, a very well developed theory of fractal dimensions) very often
continues to have mainly a qualitative character and there are no many tools for a
quantitative analysis of their behavior after execution of infinitely many steps of a
self-similarity process of construction.
Usually, we can measure fractals in a way and can give certain numerical answers
to questions regarding fractals only if a finite number of steps in the procedure of their
construction has been executed. The same questions can remain without any answer
if we consider execution of an infinite number of steps. For example, let us consider
the famous fractal construction – Cantor’s set (see Fig. 1). If a finite number of steps,
n, has been done constructing Cantor’s set, then we are able to describe numerically
the set being the result of this operation. It will have 2n intervals having the length
0
Received March 16, 2006. Revised September 12, 2006.
2000 Mathematics Subject Classification: 28A80, 40A05, 40G99, 03E99.
0
Keywords: Fractals, infinite and infinitesimal numbers, numeral systems.
0
2
Yaroslav D. Sergeyev
Step 0
Step 1
Step 2
Step 3
Step 4
F IGURE 1. Cantor’s construction.
1
3n
each. Obviously, the set obtained after n + 1 iterations will be different and we
also are able to measure the lengths of the intervals forming the second set. It will
have 2n+1 intervals having the length 3n1+1 each. The situation changes drastically in
the limit because we are not able to distinguish results of n and n + 1 steps of the
construction if n is infinite.
We also are not able to distinguish at infinity the results of the following two
processes that both use Cantor’s construction but start from different positions. The
first one is the usual Cantor’s set and it starts from the interval [0, 1], the second starts
from the couple of intervals [0, 13 ] and [ 23 , 1]. In spite of the fact that for any given
finite number of steps, n, the results of the constructions will be different for these
two processes, we have no tools to distinguish and, therefore, to measure them at
infinity.
Another class of fractal objects that defies length measurement are spirals that fascinated mathematicians throughout the ages (see, e.g., [12]). Let us consider two
kinds of polygonal spirals shown in Figs. 2 and 3. Both of them are geometric polygonal spirals related to geometric sequences. The spiral shown in Fig. 2 is constructed
as follows. The unit interval is our initial piece and we draw it vertically from bottom
to top. At the end we make a right turn and draw the unit interval again from left to
right. Then we draw the interval having a length q < 1 by continuation in the same
direction from left to right. At the end we make another right turn and draw again the
same interval having the length q from top to bottom. At the end of this line we draw
the interval with the length q2 and continue using the same principle. Fig. 3 shows
the same construction for q > 1. Evidently, for q = 1 we obtain just a square.
If we try to calculate the length of geometric polygonal spirals, we obtain immediately that it is equal to
∞
S = 2(1 + q + q2 + q3 + . . .) = 2 ∑ qi ,
(1.1)
i=0
which is a geometric series and, therefore, its limiting length for q < 1 is 2/(1 − q),
i.e., a finite value different for each given q. Then, for all q > 1 traditional analysis
Measuring fractals by infinite and infinitesimal numbers
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F IGURE 2. The first construction steps of a polygonal geometric spiral with q < 1.
F IGURE 3. The first construction steps of a polygonal geometric spiral with q > 1.
tells us that the spiral has the infinite length, i.e., we are not able to distinguish the
spirals in dependence of the value of q.
In this paper, we show how a recently developed approach (see [8, 15, 16, 17, 18,
19]) that allows one to write down infinite and infinitesimal numbers and to execute
arithmetical operations with them can be used for measuring fractals at infinity. Particularly, the lengths of intervals of Cantor’s set and the lengths of spirals from Figs. 2
and 3 for any q will be calculated.
The rest of the paper is organized as follows. Section 2 introduces the new methodology and Section 3 describes a general framework allowing one to express by a finite
number of symbols not only finite but infinite and infinitesimal numbers, too. Section 4 describes how infinite and infinitesimal numbers can be used for measuring
fractal objects. Finally, Section 5 contains a brief conclusion.
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Yaroslav D. Sergeyev
2. M ETHODOLOGY
Usually, when mathematicians deal with infinite objects (sets or processes) it is
supposed that human beings are able to execute certain operations infinitely many
times (see [1, 2, 3, 10, 14]). For example, in a fixed numeral system it is possible
to write down a numeral1 with any number of digits. However, this supposition is
an abstraction (courageously declared by constructivists in [9]) because we live in
a finite world and all human beings and/or computers finish operations they have
started.
The new computational paradigm introduced in [16, 17, 18, 19] does not use this
abstraction and, therefore, is closer to the world of practical calculations than traditional approaches. Its strong computational character is enforced also by the fact
that the first simulator of the Infinity Computer able to work with infinite, finite, and
infinitesimal numbers introduced in [16, 17, 18, 19] has been already realized (see
[8, 15]).
In order to introduce the new methodology, let us consider a study published in
Science by Peter Gordon (see [6]) where he describes a primitive tribe living in Amazonia - Pirahã - that uses a very simple numeral system for counting: one, two, many.
For Pirahã, all quantities bigger than two are just ‘many’ and such operations as 2+2
and 2+1 give the same result, i.e., ‘many’. Using their weak numeral system Pirah ã
are not able to see, for instance, numbers 3, 4, 5, and 6, to execute arithmetical operations with them, and, in general, to say anything about these numbers because in
their language there are neither words nor concepts for that. Moreover, the weakness
of their numeral system leads to such results as
‘many’ + 1 = ‘many’,
‘many’ + 2 = ‘many’,
which are very familiar to us in the context of views on infinity used in the traditional
calculus
∞ + 1 = ∞,
∞ + 2 = ∞.
This observation leads us to the following idea: Probably our difficulty in working
with infinity is not connected to the nature of infinity but is a result of inadequate
numeral systems used to express numbers.
We start by introducing three postulates that will fix our methodological positions
with respect to infinite and infinitesimal quantities and mathematics, in general.
Postulate 1. We accept that human beings and machines are able to execute only
a finite number of operations.
1We remind that numeral is a symbol or group of symbols that represents a number. The difference
between numerals and numbers is the same as the difference between words and the things they refer
to. A number is a concept that a numeral expresses. The same number can be represented by different
numerals. For example, the symbols ‘6’, ‘six’, and ‘VI’ are different numerals, but they all represent
the same number.
Measuring fractals by infinite and infinitesimal numbers
5
Thus, we accept that we shall never be able to give a complete description of
infinite processes and sets due to our finite capabilities. Particularly, this means that
we accept that we are able to write down only a finite number of symbols to express
numbers.
The second postulate that will be adopted is due to the following consideration.
In natural sciences, researchers use tools to describe the object of their study and the
used instrument influences results of observations. When physicists see a black dot
in their microscope they cannot say: The object of observation is the black dot. They
are obliged to say: the lens used in the microscope allows us to see the black dot and
it is not possible to say anything more about the nature of the object of observation
until we’ll not change the instrument - the lens or the microscope itself - by a more
precise one.
Due to Postulate 1, the same happens in mathematics studying natural phenomena, numbers, and objects that can be constructed by using numbers. Numeral systems used to express numbers are among the instruments of observations used by
mathematicians. Usage of powerful numeral systems gives possibility to obtain more
precise results in mathematics in the same way as usage of a good microscope gives
a possibility to obtain more precise results in physics. However, the capabilities of
all mathematical tools will be always limited due to Postulate 1.
Postulate 2. Following natural sciences, we shall not tell what are the mathematical objects we deal with; we just shall construct more powerful tools that will allow
us to improve our capacities to observe and to describe properties of mathematical
objects.
Particularly, this means that from our point of view, axiomatic systems do not
define mathematical objects but just determine formal rules for operating with certain
numerals reflecting some properties of the studied mathematical objects.
After all, we want to treat infinite and infinitesimal numbers in the same manner
as we are used to deal with finite ones, i.e., by applying the philosophical principle
of Ancient Greeks ‘The part is less than the whole’. This principle, in our opinion,
very well reflects organization of the world around us but is not incorporated in many
traditional infinity theories where it is true only for finite numbers.
Postulate 3. Following Ancient Greeks, we adopt the principle ‘The part is less
than the whole’ to all numbers (finite, infinite, and infinitesimal) and to all sets and
processes (finite and infinite).
Due to this declared applied statement, such concepts as bijection, numerable and
continuum sets, cardinal and ordinal numbers cannot be used in this paper because
they belong to the theories working with different assumptions 2. However, the approach proposed here does not contradict Cantor. In contrast, it evolves his deep
ideas regarding existence of different infinite numbers in a more applied way.
2
As a consequence, the approach used in this paper is different also with respect to non-standard
analysis introduced in [14] and built using Cantor’s ideas.
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Yaroslav D. Sergeyev
Let us start our consideration by studying situations arising in practice when it is
necessary to operate with extremely large quantities (see [16] for a detailed discussion). Imagine that we are in a granary and the owner asks us to count how much
grain he has inside it. There are a few possibilities of finding an answer to this question. The first one is to count the grain seed by seed. Of course, nobody can do this
because the number of seeds is enormous.
To overcome this difficulty, people take sacks, fill them in with seeds, and count
the number of sacks. It is important that nobody counts the number of seeds in a sack.
At the end of the counting procedure, we shall have a number of sacks completely
filled and some remaining seeds that are not sufficient to complete the next sack. At
this moment it is possible to return to the seeds and to count the number of remaining
seeds that have not been put in sacks (or a number of seeds that it is necessary to add
to obtain the last completely full sack).
If the granary is huge and it becomes difficult to count the sacks, then trucks or
even big train waggons are used. Of course, we suppose that all sacks contain the
same number of seeds, all trucks – the same number of sacks, and all waggons – the
same number of trucks. At the end of the counting we obtain a result in the following
form: the granary contains 17 waggons, 23 trucks, 2 sacks, and 84 seeds of grain.
Note, that if we add, for example, one seed to the granary, we can count it and see
that the granary has more grain. If we take out one waggon, we again be able to say
how much grain has been subtracted.
Thus, in our example it is necessary to count large quantities. They are finite but
it is impossible to count them directly using elementary units of measure, u 0 , i.e.,
seeds, because the quantities expressed in these units would be too large. Therefore,
people are forced to behave as if the quantities were infinite.
To solve the problem of ‘infinite’ quantities, new units of measure, u 1 , u2 , and u3 ,
are introduced (units u1 – sacks, u2 – trucks, and u3 – waggons). The new units
have the following important peculiarity: it is not known how many units u i there are
in the unit ui+1 (we do not count how many seeds are in a sack, we just complete
the sack). Every unit ui+1 is filled in completely by the units ui . Thus, we know
that all the units ui+1 contain a certain number Ki of units ui but this number, Ki , is
unknown. Naturally, it is supposed that Ki is the same for all instances of the units.
Thus, numbers that it was impossible to express using only initial units of measure
are perfectly expressible if new units are introduced. This key idea of counting by
introduction of new units of measure will be used in the paper to deal with infinite
quantities.
In order to have a possibility to write down infinite and infinitesimal numbers by
a finite number of symbols, we need at least one new numeral expressing an infinite
Measuring fractals by infinite and infinitesimal numbers
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(or an infinitesimal) number corresponding to the chosen infinite unit of measure 3.
Then, it is necessary to propose a new numeral system fixing rules for writing down
infinite and infinitesimal numerals and to describe arithmetical operations with them.
3. I NFINITE AND INFINITESIMAL NUMBERS AND OPERATIONS WITH
THEM
Different numeral systems have been developed by humanity to describe finite
numbers. More powerful numeral systems allow us to write down more numerals
and, therefore, to express more numbers. A new positional numeral system with
infinite radix described in this section evolves the idea of separate count of units with
different exponents used in traditional positional systems to the case of infinite and
infinitesimal numbers.
The infinite radix of the new system is introduced as the number of elements of the
set N of natural numbers expressed by the numeral ① called grossone. This mathematical object is introduced by describing its properties postulated by the Infinite
Unit Axiom consisting of three parts: Infinity, Identity, and Divisibility (we introduce
them soon). This axiom is added to axioms for real numbers similarly to addition
of the axiom determining zero to axioms of natural numbers when integer numbers
are introduced. This means that it is postulated that associative and commutative
properties of multiplication and addition, distributive property of multiplication over
addition, existence of inverse elements with respect to addition and multiplication
hold for grossone as for finite numbers.
Note that usage of a numeral indicating totality of the elements we deal with is not
new in mathematics. It is sufficient to remind the theory of probability where events
can be defined in two ways. First, as union of elementary events; second, as a sample space, Ω, of all possible elementary events from where some elementary events
have been excluded. Naturally, the second way to define events becomes particularly
useful when the sample space consists of infinitely many elementary events.
The Infinite Unit Axiom consists of the following three statements:
Infinity: For any finite natural number n it follows n < ①.
Identity: The following relations link ① to identity elements 0 and 1
0 · ① = ① · 0 = 0, ① − ① = 0,
①
= 1, ① 0 = 1, 1 ① = 1, 0 ① = 0.
①
(3.1)
Divisibility: For any finite natural number n sets Nk,n , 1 ≤ k ≤ n, being the nth
parts of the set, N, of natural numbers have the same number of elements
3Note that introduction of a new numeral for expressing infinite and infinitesimal numbers is similar
to introduction of the concept of zero and the numeral ‘0’ that in the past have allowed people to develop
positional systems being more powerful than numeral systems existing before.
Yaroslav D. Sergeyev
8
indicated by the numeral ①
n where
Nk,n = {k, k + n, k + 2n, k + 3n, . . .},
1 ≤ k ≤ n,
n
[
Nk,n = N.
(3.2)
k=1
Divisibility is based on Postulate 3. Let us illustrate it by three examples. If we take
n = 1, then N1,1 = N and Divisibility tells that the set, N, of natural numbers has ①
elements. If n = 2, we have two sets N1,2 and N2,2 and they have ①
2 elements each.
①
elements each.
If n = 3, then we have three sets N , N , and N having
1,3
①→
N=
2,3
3,3
3
{1, 2, 3, 4, 5, 6, 7, . . . }
3,
5,
7, . . . }
N1,2 = {1,
① %
2 & N ={
2,
4,
6,
... }
2,2
N1,3 = {1,
4,
7, . . . }
%
①
→ N2,3 = {
2,
5,
... }
3 &
3,
6,
... }
N3,3 = {
Before the introduction of the new positional system let us study some properties
of grossone. First of all, as was already mentioned above, it is necessary to remind
that ① is not either Cantor’s ℵ0 or ω that have been introduced in Cantor’s theory
on the basis of different assumptions. It will be shown hereinafter that grossone
unifies both cardinal and ordinal aspects in the same way as finite numerals unify
them. Its role in our infinite arithmetic is similar to the role of the number 1 in the
finite arithmetic and it will serve us as the basis for construction of other infinite and
infinitesimal numbers.
We start by the following important comment: to introduce ①
n we do not try to
count elements k, k + n, k + 2n, k + 3n, . . . In fact, we cannot do this due to the accepted
Postulate 1. In contrast, we apply Postulate 3 and state that the number of elements
of the nth part of the set, i.e., ①
n , is n times less than the number of elements of the
whole set, i.e., than ①. In terms of our granary example ① can be interpreted as the
number of seeds in the sack. Then, if the sack contains ① seeds, its nth part contains
① seeds. It is worthy to emphasize that, since the numbers ① have been introduced
n
n
as numbers of elements of sets Nk,n , they are integer.
The introduced numerals ①
n and the sets Nk,n allow us immediately to calculate
the number of elements of certain infinite sets. For example, due to the introduced
axiom, the set
{3, 8, 13, 18, 23, 28, . . .} = N3,5
Measuring fractals by infinite and infinitesimal numbers
9
and, therefore, has ①
5 elements. The number of elements of sets being union, intersection, difference, or product of other sets of the type N k,n is defined in the same
way as these operations are defined for finite sets. Thus, we can define the number of
elements of sets being results of these operations with finite sets and infinite sets of
the type Nk,n . For example, the set
{3, 8, 13, 18, 23, 28, . . .}\{3, 23} = N3,5 \{3, 23}
and, therefore, it has ①
5 − 2 elements.
Other results regarding calculating the number of elements of infinite sets can be
found in [16, 19]. Particularly, it is shown that the number of elements of the set, Z,
of integers is equal to 2① + 1 and the number of elements of the set, Q, of different
rational numerals is equal to 2①2 + 1.
The new numeral ① allows us to write down the set, N, of natural numbers in the
form
N = {1, 2, 3, . . . ① − 2, ① − 1, ①}
(3.3)
because grossone has been introduced as the number of elements of the set of natural
numbers (similarly, the number 3 is the number of elements of the set {1, 2, 3}).
Thus, grossone is the biggest natural number and infinite numbers
. . . ① − 3, ① − 2, ① − 1
(3.4)
less than grossone are also natural numbers as the numbers 1, 2, 3, . . . They can be
viewed both in terms of sets of numbers and in terms of grain. For example, ① − 1
can be interpreted as the number of elements of the set N from which a number has
been excluded. In terms of our granary example ① − 1 can be interpreted as a sack
minus one seed.
Note that the set (3.3) is the same set of natural numbers we are used to deal with.
Infinite numbers (3.4) also take part of the usual set, N, of natural numbers 4. The
difficulty to accept existence of infinite natural numbers is in the fact that traditional
numeral systems did not allow us to see them. In the same way as Pirahã are not able
to see, for instance, numbers 3, 4, and 5 using their weak numeral system, traditional
numeral systems did not allow us to see infinite natural numbers that we can see now
using the new numeral ①.
Postulate 3 and the Infinite Unit Axiom allow us to obtain the following important
result: the set N is not a monoid under addition. In fact, the operation ① + 1 gives us
as the result a number grater than ①. Thus, by definition of grossone, ① + 1 does not
belong to N and, therefore, N is not closed under addition and is not a monoid.
This result also means that adding the Infinite Unit Axiom to the axioms of natural
b and including N
numbers defines the set of extended natural numbers indicated as N
4This point is one of the differences with respect to non-standard analysis (see [13, 14]) where
infinite numbers are not included in N.
Yaroslav D. Sergeyev
10
as a proper subset
b = {1, 2, . . . , ① − 1, ①, ① + 1, . . . , ①2 − 1, ①2 , ①2 + 1, . . .}.
N
Again, extended natural numbers grater than grossone can also be interpreted in the
terms of sets of numbers. For example, ① + 3 as the number of elements of the set
N ∪ {a, b, c} where numbers a, b, c ∈
/ N and ①2 as the number of elements of the set
C = {(a1 , a2 ) : a1 ∈ N, a2 ∈ N}.
In terms of our granary example ① + 3 can be interpreted as one sack plus three seeds
and ①2 as a truck.
b , of extended integer numbers
Analogously, we can consider the set, Z
b = {. . . , −① − 1, −①, −① + 1 . . . , −2, −1, 0, 1, 2, . . ., ① − 1, ①, ① + 1, . . .}.
Z
b and Z
b ? Our
What can we say now about the number of elements of the sets N
positional numeral system with the radix ① dose not allow us to say anything because
it does not contain numerals able to express such numbers (see Postulates 1 and 2).
It is necessary to introduce in a way a more powerful numeral system defining new
numerals ②, ③, etc. However, in spite of the fact that the numeral system using
b and Z
b , we can
grossone does not allow us to express the numbers of elements of N
b
b
work with those subsets of N and Z that can be defined by using numerals written
down in our positional numeral system with the radix ①.
We have already started to write down simple infinite numbers and to execute arithmetical operations with them without concentrating our attention upon this question.
In general, to express a number C in the new numeral positional system with base ①
we subdivide C into groups corresponding to powers of ①:
C = c p m ① p m + . . . + c p 1 ① p 1 + c p 0 ① p 0 + c p −1 ① p −1 + . . . + c p −k ① p −k .
(3.5)
Then, the record
C = c p m ① p m . . . c p 1 ① p 1 c p 0 ① p 0 c p −1 ① p −1 . . . c p −k ① p −k
(3.6)
represents the number C, where finite numbers ci are called infinite grossdigits and
can be both positive and negative; numbers pi are called grosspowers and can be
finite, infinite, and infinitesimal (the introduction of infinitesimal numbers will be
given soon). The numbers pi are such that pi > 0, p0 = 0, p−i < 0 and
pm > pm−1 > . . . p2 > p1 > p−1 > p−2 > . . . p−(k−1) > p−k .
In the traditional positional systems there exists a convention that a digit a i shows how
many powers bi are present in the number and the radix b is not written explicitly. In
the record (3.6) we write ① pi explicitly because in the new numeral positional system
the number i in general is not equal to the grosspower pi . This gives possibility to
84 −3.1
write, for example, such numbers as 37 ①4 19
①
where p1 = 4, p−1 = −3.1. Grossdigits ci , −k ≤ i ≤ m, can be integer or fractional and expressed by many symbols; in
our example, c4 = 73 and c−3.1 = 84
19 .
Measuring fractals by infinite and infinitesimal numbers
11
Finite numbers in this new numeral system are represented by numerals having
only one grosspower equal to zero. In fact, if we have a number C such that m = k = 0
in representation (3.6), then due to (3.1) we have C = c 0 ①0 = c0 . Thus, the number C
in this case does not contain infinite units and is equal to the grossdigit c 0 which being
a conventional finite number can be expressed by any positional system with finite
base b (or by another traditional numeral system). It is important to emphasize that
the grossdigit c0 can be integer or fractional and can be expressed by a few symbols
in contrast to the traditional positional systems where each digit is integer and is
represented by one symbol from the alphabet {0, 1, 2, . . . , b − 1}. Thus, the grossdigit
c0 shows how many finite units and/or parts of the finite unit, 1 = ① 0 , there are in the
number C.
Infinite numbers in this numeral system are expressed by numerals having at least
one grosspower grater than zero. In the following example the left-hand expression
presents the way to write down infinite numbers and the right-hand shows how the
value of the number is calculated:
21.4①23 -1.45①3.4 852.1①−66.2 = 21.4①23 − 1.45①3.4 + 852.1①−66.2 .
If a grossdigit c pi is equal to 1 then we write ① pi instead of 1① pi . Analogously,
if power ①0 is the lowest in a number then we often use simply the corresponding
grossdigit c0 without ①0 , for instance, we write 23①14 5 instead of 23①14 5①0 or 8
instead of 8①0 .
Infinitesimal numbers are represented by numerals having only negative grosspowers. The simplest number from this group is ①−1 = 1 being the inverse element
①
with respect to multiplication for ①:
1
1
· ① = ① · = 1.
①
①
(3.7)
Note that all infinitesimals are not equal to zero. Particularly, 1 > 0 because 1 > 0
①
and ① > 0. It has a clear interpretation in our granary example. Namely, if we have
a sack and it contains ① seeds then one sack divided by ① is equal to one seed. Vice
versa, one seed, i.e., 1 , multiplied by the number of seeds in the sack, ①, gives one
①
sack of seeds.
Let us now introduce arithmetical operations for infinite, infinitesimal, and finite
numbers (see [16] for a detailed discussion and examples). The numbers A, B, and
their sum C are represented in the record of the type
K
A = ∑ a ki ① ki ,
i=1
M
B=
∑ bm ①m ,
j
j
j=1
L
C = ∑ c li ① li .
(3.8)
i=1
The operation of addition of two given infinite numbers A and B returns as the
result an infinite number C constructed as follows (the operation of subtraction is
a direct consequence of that of addition and is thus omitted). Then the result C is
Yaroslav D. Sergeyev
12
constructed by including in it all items aki ①ki from A such that ki 6= m j , 1 ≤ j ≤ M ,
and all items bm j ①m j from B such that m j 6= ki , 1 ≤ i ≤ K. If in A and B there are items
such that ki = m j for some i and j then this grosspower ki is included in C with the
grossdigit bki + aki , i.e., as (bki + aki )①ki . It can be seen from this definition that the
introduced operation enjoys the usual properties of commutativity and associativity
due to definition of grossdigits and the fact that addition for each grosspower of ① is
executed separately.
The operation of multiplication of two given infinite numbers A and B from (3.8)
returns as the result the infinite number C constructed as follows.
M
C=
∑ C j,
j=1
K
C j = b m j ① m j · A = ∑ a k i b m j ① k i +m j ,
1 ≤ j ≤ M.
(3.9)
i=1
Similarly to addition, the introduced multiplication is commutative and associative.
It is easy to show that the distributive property is also valid for these operations.
In the operation of division of a given infinite number C by an infinite number B
we obtain an infinite number A and a reminder R that can be also equal to zero, i.e.,
C = A · B + R.
The number A is constructed as follows. The numbers B and C are represented in
the form (3.8). The first grossdigit akK and the corresponding maximal exponent kK
are established from the equalities
a k K = c l L /b m M ,
kK = lL − m M .
(3.10)
Then the first partial reminder R1 is calculated as
R1 = C − akK ①kK · B.
(3.11)
If R1 6= 0 then the number C is substituted by R1 and the process is repeated by a complete analogy. The grossdigit akK−i , the corresponding grosspower kK −i and the partial
reminder Ri+1 are computed by formulae (3.12) and (3.13) obtained from (3.10) and
(3.11) as follows: lL and clL are substituted by the highest grosspower ni and the
corresponding grossdigit rni of the partial reminder Ri that in its turn substitutes C:
a k K −i = r n i /b m M ,
k K −i = n i − m M .
(3.12)
Ri+1 = Ri − akK−i ①kK−i · B, i ≥ 1.
(3.13)
The process stops when a partial reminder equal to zero is found (this means that the
final reminder R = 0) or when a required accuracy of the result is reached.
4. M EASURING OBJECTS BEING RESULTS OF SELF - SIMILARITY PROCESSES AT
INFINITY
We start by proving the following important result: the number of elements of any
infinite sequence is less or equal to ①. To demonstrate this we need to recall the
definition of the infinite sequence: ‘An infinite sequence {a n }, an ∈ A for all n ∈ N, is
Measuring fractals by infinite and infinitesimal numbers
13
a function having as the domain the set of natural numbers, N, and as the codomain
a set A ⊆ R’.
We have postulated in the Infinite Unit Axiom that the set N has ① elements. Thus,
due to the sequence definition given above, any sequence having N as the domain has
① elements.
One of the immediate consequences of the understanding of this result is that any
b,
process can have at maximum ① elements5. For example, if we consider the set, Z
of extended integer numbers then starting from the number 1, it is possible to arrive
at maximum to ①
. . . , −3, −2, −1, 0, 1, 2, 3, 4, . . . ① − 2, ① − 1, ①, ① + 1, ① + 2, ① + 3, . . .
|
{z
}
①
Starting from 0 it is possible to arrive at maximum to ① − 1
(4.1)
. . . , −3, −2, −1, 0, 1, 2, 3, 4, . . . ① − 2, ① − 1, ①, ① + 1, ① + 2, ① + 3, . . .
|
{z
}
①
Starting from -1 it is possible to to arrive at maximum to ① − 2
(4.2)
. . . , −3, −2, −1, 0, 1, 2, 3, 4, . . . ① − 2, ① − 1, ①, ① + 1, ① + 2, ① + 3, . . . (4.3)
|
{z
}
①
Of course, since we have postulated that our possibilities to express numerals are
finite, it depends on the chosen numeral system which numbers among ① members
of these processes we can observe.
In order to be able to measure fractals at infinity, we should reconsider the theory
of divergent series from the new viewpoint introduced in the previous sections. The
introduced numeral system allows us to express not only different finite numbers but
also different infinite numbers. Therefore, due to Postulate 3, we should explicitly
indicate the number of items in all sums independently on the fact whether this number is finite or infinite. Due to Postulate 2, we shall be able to calculate the sum if its
items, the number of items, and the result are expressible in the numeral system used
for calculations. It is important to notice that even though a sequence cannot have
more than ① elements, the number of items in a sum can be greater than grossone
because the process of summing up not necessary should be executed by a sequential
adding items.
For instance, let us consider two infinite series
S1 = 2 + 2 + 2 + 2 + 2 + . . .
S2 = 1 + 2 + 1 + 2 + 1 + 2 + 1 . . .
5This observation has a deep relation to the Axiom of Choice. The Infinite Unit Axiom postulates
that any process can have at maximum ① elements, thus the process of choice too and, as a consequence,
it is not possible to choose more than ① elements from a set. This observation also emphasizes the fact
that the parallel computational paradigm is significantly different with respect to the sequential one
because p parallel processes can choose p① elements from a set.
Yaroslav D. Sergeyev
14
The traditional analysis gives us a very poor answer that both of them diverge to
infinity. Such operations as S1 − S2 or SS12 are not defined. From the new point of
view, the sums S1 and S2 can be calculated but it is necessary to indicate explicitly
the number of items in both sums.
Suppose that the sum S1 has m items and the sum S2 has n items:
S1 (m) = 2| + 2 + 2{z+ . . . + 2},
m
Then S1 (m) = 2m and
S2 (n) = 1| + 2 + 1 + 2 +
{z1 + 2 + 1 + . .}. .
S2 (n) = 1| + 2 + 1 + 2 +
{z1 + 2 + 1 + . .}. =
n
n
k + 2k = 3k,
if n = 2k,
k + 2k + 1 = 3k + 1, if n = 2k + 1,
and giving numerical values (finite or infinite) to m and n we obtain numerical values
for results of the sums. If, for instance, m = n = 3① then we obtain S 1 (3①) = 6①,
S2 (3①) = 4.5① because ① is even (since, due to the Infinite Unit Axiom, ①
2 is integer) and
S2 (3①) − S1 (3①) = −1.5① < 0.
If m = ① and n = 3① + 1 we obtain S1 (①) = 2①, S2 (3① + 1) = 4.5① + 1 and it
follows
S2 (3① + 1) − S1 (①) = 2.5① + 1 > 0.
If m = 3① and n = 4① we obtain S1 (3①) = 6①, S2 (4①) = 6① and it follows
S2 (4①) − S1 (3①) = 0.
Analogously, the expression SS21 ((nk)) can be calculated.
Let us return now to Cantor’s construction and remind that if a finite number of
steps, n, has been executed in Cantor’s construction starting from the interval [0, 1]
then we are able to describe numerically the set being the result of this operation. It
will have 2n intervals having the length 31n each. Obviously, the set obtained after
n + 1 iterations will be different and we also are able to measure the lengths of the
intervals forming the second set. It will have 2n+1 intervals having the length 3n1+1
each. The situation changes drastically in the limit because traditional approaches
are not able to distinguish results of n and n + 1 steps of the construction if n is
infinite. Now, we can do it using the introduced infinite and infinitesimal numbers.
Since the construction of Cantor’s set is a process, it cannot contain more then ①
steps (see discussion related to the example (4.1)-(4.3)). Thus, if we start the process
from the interval [0, 1], after ① steps Cantor’s set consists of 2① intervals and their
total length, Ln , is expressed in infinitesimals: L(①) = ( 23 )① , i.e., the set has a well
defined infinite number of intervals and each of them has the infinitesimal length
equal to 3−① . Analogously, after ① − 1 steps Cantor’s set consists of 2①−1 intervals
and their total length is expressed in infinitesimals: L(①) = ( 23 )①−1 . Thus, the length
Measuring fractals by infinite and infinitesimal numbers
15
Ln for any (finite or infinite) number of steps, n, where 1 ≤ n ≤ ① and is expressible
in the chosen numeral system can be calculated.
It is important to notice here that (again due to the limitation illustrated by the
example (4.1)-(4.3)) it is not possible to count one by one all the intervals at Cantor’s
set if their number is superior to ①. For instance, after ① steps it has 2 ① intervals
and they cannot be counted one by one because 2 ① > ① and any process (including
that of the sequential counting) cannot have more that ① steps.
Let us consider now two processes that both use Cantor’s construction but start
from different initial conditions. Traditional approaches do not allow us to distinguish them at infinity in spite of the fact that for any given finite number of steps,
n, the results of the constructions are different and can be calculated. Using the new
approach we are able to study the processes numerically also at infinity. For example,
if the first process is the usual Cantor’s set and it starts from the interval [0, 1] and
the second one starts from the couple of intervals [0, 13 ] and [ 23 , 1] then after ①
2 steps
①
the result of the first process will be the set consisting of 2 2 intervals and its length
①
①
L( ① ) = ( 2 ) 2 . The second set after ① steps will consists of 2 2 +1 intervals and its
2
3
2
①
length L( ① + 1) = ( 2 ) 2 +1 . Naturally, it becomes possible to measure by a complete
2
3
analogy other classical fractals such as the Koch Curve, the Sierpinski Carpet, etc.
In order to be able to calculate lengths of geometric polygonal spirals from Figs. 2
i
and 3 at infinity, we need to consider (see (1.1)) the geometric series ∑∞
i=0 q from the
new viewpoint. Traditional analysis proves that it converges to 1−1 q for q such that
−1 < q < 1. We are able to give a more precise answer for all values of q. Due to
Postulate 3, to do this we should fix the number of items in the sum. If we suppose
that it contains n items then
n
Qn = ∑ qi = 1 + q + q 2 + . . . + q n .
(4.4)
i=0
By multiplying the left hand and the right hand parts of this equality by q and by
subtracting the result from (4.4) we obtain
Qn − qQn = 1 − qn+1
and, as a consequence, for all q 6= 1 the formula
Qn =
1 − qn+1
1−q
(4.5)
holds for finite and infinite n. Thus, the possibility to express infinite and infinitesimal numbers allows us to take into account infinite n too and the value q n+1 being
infinitesimal for a finite q < 1 and infinite for q > 1. Moreover, we can calculate Q n
Yaroslav D. Sergeyev
16
for q = 1 also because in this case we have just
Qn = 1| + 1 + 1{z+ . . . + 1} = n + 1.
n+1
As the first example, let us consider a spiral with q = 13 . Traditional analysis tells us
that the length of the spiral is equal to
∞
1
2
=
= 3.
i
1 − 13
i=0 3
S = 2∑
By using the new computational paradigm we are able to give a more precise answer
because we are able to distinguish different infinite and infinitesimal numbers and,
as a consequence, we can speak not only about tendencies at infinity, as traditional
analysis does, but give precise numerical answers. Thus, if we know how many
steps, n, have been executed during the process of the construction of the spiral, we
can calculate its length Sn for finite and infinite n using formulae (1.1), (4.4), and
(4.5): Sn = 2Qn . For example, if ① − 1 steps have been executed, the spiral will have
the length
1
1 − 3①
1 1
1
1
S①−1 = 2(1 + + 2 + . . . + ①−1 ) = 2 ·
= 3 − ①−1 .
1
3 3
3
3
1− 3
(4.6)
We can see from this formula that the new numeral system allows us to distinguish
and to measure the infinitesimal part of the length of the spiral, 3①1−1 , that was invisible for traditional numeral systems. Analogously, if n = ①, the length of the spiral is
equal to
1
S ① = 2 (1 +
1 − 3①+1
1 1
1
1
1
+ 2 + . . . + ①−1 + ① ) = 2 ·
= 3− ①.
1
3 3
3
3
3
1− 3
(4.7)
The second spiral has been obtained from the first one by adding two intervals having
infinitesimal length 31① . We can obtain this value by subtracting the right part of (4.6)
from the right part of (4.7)
1
1
2
S① − S①−1 = 3 − ① − (3 − ①−1 ) = ① .
3
3
3
The new numeral system allows us to measure also the lengths of spirals with
q > 1. For example, from the traditional point of view it is not possible to measure
and to compare spirals having q = 2 and q = 3. We are forced just to say that their
lengths are infinite because the series
∞
1 + 2 + 4 + 8 + . . . = ∑ 2i ,
i=0
∞
1 + 3 + 9 + 27 + . . . = ∑ 3i
i=0
are divergent. The new approach allows us to measure both spirals and to compare
them. Suppose that the spiral with q = 2 has been constructed in n steps and the spiral
Measuring fractals by infinite and infinitesimal numbers
17
with q = 3 in m steps and we want to calculate their lengths An and Bm . It follows
from (1.1), (4.4), and (4.5) that
1 − 2n+1
1 − 3m+1
= 2 n+2 − 2 ,
Bm = 2
= 3m+1 − 1.
1−2
1−3
Both formulae work for finite and infinite values of n and m and allow us to measure
the lengths of spirals easily. For example, if n = m = ①
2 − 1, then obviously
An = 2
①
A ① −1 = 2 2 +1 − 2 ,
2
①
B ① −1 = 3 2 − 1 .
2
If in the construction of the first spiral one more step has been executed, i.e., n = ①
2 ,
①
and in the construction of the second spiral m = + 1, i.e., two steps have been
2
added, then
①
①
A ① = 2 2 +2 − 2 ,
B ① +1 = 3 2 +2 − 1 .
2
2
We can obtain the same results by direct calculation, i.e., by adding two pieces of the
①
①
①
length 2 2 to A ① −1 and two times pieces of the length 3 2 and 3 2 +1 to B ① −1
2
2
①
① +1
A ① −1 + 2 · 2 2 = 2 2
① +1
−2+2 2
① +2
=22
− 2,
2
①
①
①
①
①
①
B ① −1 + 2 (3 2 + 3 2 +1 ) = 3 2 − 1 + 2 (3 2 + 3 2 +1 ) = 3 2 +2 − 1 .
2
5. A
BRIEF CONCLUSION
It has been shown in the paper that the new powerful numeral system allowing us
to express not only finite but also infinite and infinitesimal numbers gives a lot of
new (in comparison with traditional numeral systems able to express only finite numbers) information about behavior of fractal objects at infinity and can be successfully
applied for measuring fractals.
It has been emphasized that the philosophical triad – researcher, object of investigation, and tools used to observe the object – existing in such natural sciences as
physics and chemistry exists in mathematics too. Usage of powerful numeral systems
gives a possibility to obtain more precise results in mathematics in the same way
as usage of a good microscope gives a possibility to obtain more precise results in
physics. Infinite and infinitesimal numbers introduced in [15, 16, 17, 18, 19]) allow
us to distinguish at infinity different infinite iterations and different fractal objects
corresponding to that iterations that were undistinguishable when traditional finite
numbers were used.
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18
Yaroslav D. Sergeyev
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